Last update: 26 February 2013
A. Young invented the theory of standard Young tableaux in order to describe the representation theory of the symmetric group the group of matrices such that
Young himself began to generalize the theory and in [You1929] he provided a theory for the Weyl groups of type B, i.e. the hyperoctahedral groups of matrices such that
In the same paper Young also treated the Weyl group of matrices such that
W. Specht [Spe1932] generalized the theory to cover the complex reflection groups consisting of matrices such that
In the classification [STo1954] of finite groups generated by complex reflections there is a single infinite family of groups and exactly 34 others, the “exceptional” complex reflection groups. The groups are the groups of matrices such that
Though we do not know of an early reference which generalizes the theory of Young tableaux to these groups, it is not difficult to see that the method that Young uses for the Weyl groups extends easily to handle the groups
Special cases of the groups are
The order of is Let be the matrix with 1 in the position and all other entries 0. Then can be presented by generators
where is a primitive root of unity, and relations
The group is the subgroup of index in generated by
More recently there has been an interest in Iwahori-Hecke algebras associated to reflection groups and there has been significant work generalizing the constructions of A. Young to these algebras. Iwahori-Hecke algebras of types A, B and D were handled by Hoefsmit [Hoe1974] and other aspects of the theory for these algebras were developed by Dipper, James and Murphy [DJa1987], [DJM1995], Gyoja [Gyo1986] and Wenzl [Wen1988]. In 1994, Ariki and Koike [AKo1994] introduced cyclotomic Hecke algebras for the complex reflection groups and they generalized the Young tableau theory to these algebras. Theorem 3.18 below shows that the theory of [AKo1994] is a special case of an even more general theory for affine Hecke algebras.
Let The cyclotomic Hecke algebra is the algebra over given by generators and relations
The algebra is of dimension (see [AKo1994]).
Fact (c) says that the representation theory of the groups is a special case of the representation theory of the algebras
The work of Broué, Malle and Michel [BMM1993] demonstrated that there are cyclotomic Hecke algebras associated to most complex reflection groups (even exceptional complex reflection groups). In particular, there are cyclotomic Hecke algebras corresponding to all the groups and Ariki [Ari1995] has generalized the Young tableau mechanism to these groups (see also [HRa1998]). Theorems 3.15 and the mechanism of Theorem 2.8 show that the theory of Ariki is a special case of a general construction for affine Hecke algebras.
Let be such that divides and let Let and let be a primitive root of 1. For define
i.e., are chosen so that
The algebra is the subalgebra of generated by the elements
There are three common ways of depicting affine braids [Cri1997], [GLa1997], [Jon1994]:
See Figure 1. The multiplication is by placing one cylinder on top of another, placing one annulus inside another, or placing one flagpole braid on top of another. These are equivalent formulations: an annulus can be made into a cylinder by turning up the edges, and a cylindrical braid can be made into a flagpole braid by putting a flagpole down the middle of the cylinder and pushing the pole over to the left so that the strings begin and end to its right.
The group formed by the affine braids with strands is the affine braid group of type A. The affine braid group is presented by generators and (see Figure 2) with relations
Inductively define by
By drawing pictures of the corresponding affine braids one can check that the all commute with each other. View the symbols as a basis of so that
The affine braid group contains a large abelian subgroup
where for The pole winding number of an affine braid is where is the group homomorphism defined by and The affine braid group is the subgroup of of affine braids with pole winding number equal to 0 (mod
for each nonnegative integer The lattice is a lattice of index in Then
and the group is an abelian subgroup of
The affine Hecke algebra (resp. is the quotient of the group algebra (resp. by the relations
Let and be as in (1.5) and (1.6). The subalgebra
is a commutative subalgebra of (resp. The symmetric group acts on the lattice by permuting the and the lattices are sublattices of Let and For and (resp
as elements of (resp
For each element define if is an expression of as a product of simple reflections such that is minimal. The element does not depend on the choice of the reduced expression of [Bou1968, Ch. IV §2 Ex. 23]. The sets
are bases of and respectively [Lus1989]. The center of is
and is the center of (see Theorem 4.12 below).
This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.
Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).